Sameness, Invariance, and the Structure of Representation

This paper proposes a theory of representation centered on invariance as the constitutive condition of objectivity. Any objective representational system functions as a sieve: it stabilizes identity by enforcing invariance over a structured possibility space. I demonstrate that binary relational representation under global invariance forces exactly four exhaustive stability profiles, determined by whether each argument position behaves as instance-like or role-like under substitution. These four profiles appear across representational systems: as fundamental verbs in natural language (IS, HAS, MEANS, CAUSES), as arithmetic operations in mathematics (addition, multiplication, subtraction, division), and as interaction roles in physics. This convergence is not analogical but necessary: the four-fold emerges from the constraint structure of objective binary relational representation itself.